It depends on the company.
When I was in college, I had checking & savings with Suntrust, and they wouldn't give me even a low-limit CC for gas and stuff. They promptly lost the checking & savings business. Discover gave me my first card, no problem! (I really wish there was a :confused: smiley!)
The checking & savings got moved to a local bank, and I believe it was a good help with getting my first car loan, my first house, and refi'ing my HELOC. I've never tried getting a CC from them, though.
USAA told me plainly that the more products you had with them, the better their rates and the more likely you are to be approved- the lady strongly suggested opening a checking account with them before refinancing our truck loan.
-MsMS